How to Use the Double and Half Angle Formulas for Trigonometry (Precalculus - Trigonometry 28)

In this video, we'll build on what was covered previously by using double and half-angle formulas for sine, cosine, and tangent. These formulas are not difficult; they are simply mathematical tools we need to practice. Key challenges include determining sine and cosine for unknown angles and finding the quadrant for half-angles. We start by using given information about an angle, such as cosine theta being 3/5 and theta being in the first quadrant, to determine other trigonometric functions. The first quadrant simplifies things since all trigonometric functions are positive there. Through examples, we draw triangles, apply the Pythagorean theorem, and calculate sine and tangent based on the given cosine value.

This section focuses on applying double-angle and half-angle formulas. For sine of 2 theta, we use the formula 2 sin theta cos theta. Direct calculation of sine of 2 theta by doubling the angle and figuring out trigonometry is complicated, so using the formula is more practical. We evaluate sine and cosine for theta, then use them to calculate sine of 2 theta, resulting in 24/25. For cosine of 2 theta, we use one of the three possible formulas, choosing based on the information we have. For instance, we use 2 cos^2 theta - 1 when cosine theta is given. This leads to a consistent result, showing that multiple methods yield the same answer.

When using half-angle formulas, the quadrant in which the original angle lies is crucial. We calculate sine and cosine for half the angle using specific formulas, such as sin(theta/2) = ±√((1 - cos(theta))/2). Given the original angle's quadrant, we determine the sign of the result. For instance, with theta in the first quadrant, theta/2 remains in the first quadrant, making sine and cosine positive. These principles help us correctly apply the half-angle formulas and identify whether the results are positive or negative based on the quadrant.

Further explanations on determining the quadrant for half-angles and simplifying the resulting expressions are provided. By dividing the range of the original angle by two, we find the new quadrant for theta/2. This method confirms that if theta is in the first quadrant, theta/2 is also in the first quadrant. Knowing the quadrant helps us decide the sign of sine and cosine. We use the same process for tangent of double angles and half angles, applying the appropriate formulas and ensuring the results align with the trigonometric functions' signs in their respective quadrants.

We delve into calculating the tangent for double and half angles. The formula for tangent of 2 theta is 2 tan(theta) / (1 - tan^2(theta)). Substituting known values, we simplify to get the result, ensuring signs are correct based on the quadrant. The double angle automatically adjusts signs, while the half-angle requires careful attention to the quadrant. For tangent of theta/2, we use the formula ±√((1 - cos(theta))/(1 + cos(theta))), ensuring the correct sign based on the quadrant. This comprehensive method ensures accurate results for trigonometric functions involving double and half angles.

This segment explains handling angles in different quadrants and finding corresponding trigonometric values. Given cosine and sine values and their signs, we determine the quadrant. For example, if both sine and cosine are negative, the angle is in the third quadrant. We then use the Pythagorean theorem to find missing values, ensuring the signs are correct for the given quadrant. This method is essential for accurate trigonometric calculations across different quadrants.

We continue solving for trigonometric functions given different angle conditions. Using specific formulas and the Pythagorean theorem, we calculate sine, cosine, and tangent. We also discuss how changing the angle affects the quadrant and the corresponding trigonometric values. This method provides a clear approach to solving complex trigonometric equations, ensuring accurate results.

This section focuses on simplifying complex trigonometric expressions involving double and half angles. Using appropriate formulas, we break down the expressions step by step, ensuring accurate results. By carefully considering the quadrant and applying the correct trigonometric identities, we simplify and solve these expressions effectively.

We address the challenge of finding trigonometric values for angles not on the unit circle. By rewriting angles using double or half-angle formulas, we transform difficult angles into more manageable ones. For example, rewriting 22.5 degrees as half of 45 degrees allows us to use known trigonometric values and formulas, ensuring accurate calculations.

We explore using half-angle formulas for tangent, demonstrating how to handle angles like 9 pi over 8 by reducing them to more familiar angles within the unit circle's range. This approach ensures we can accurately determine trigonometric values for complex angles, leveraging known formulas and identities.

We demonstrate proving trigonometric identities by starting with the more complex side and simplifying it using known identities. This method involves recognizing patterns, applying appropriate formulas, and confirming that both sides of the identity match. This process is crucial for mastering trigonometric proofs.

We solve trigonometric equations by identifying the appropriate identities to simplify the expressions. For example, using double-angle formulas to transform equations into solvable forms. This method involves careful selection of identities, factoring, and solving for the variable, ensuring accurate results.

We emphasize the importance of selecting the right trigonometric formulas for simplification. By choosing identities that match the given expressions, we streamline the solving process. This approach is essential for efficiently tackling complex trigonometric equations and achieving accurate solutions.

We demonstrate how to handle equations involving double angles by converting them into single-angle expressions using appropriate identities. This step-by-step process involves careful application of formulas, factoring, and solving for the desired variable, ensuring accurate and manageable solutions.

This section focuses on solving trigonometric equations through factoring. By transforming complex expressions into factorable forms and applying zero-product properties, we solve for the variable. This method is crucial for efficiently solving trigonometric equations and finding all possible solutions.

We wrap up by discussing the overall process of handling double and half angles in trigonometric equations. By using appropriate identities, factoring, and careful consideration of quadrants, we ensure accurate solutions. This comprehensive approach is essential for mastering trigonometric calculations and solving complex equations.